Contents

Definition

Given a simplicial group $\mathcal{G} \in Groups(sSet)$, hence a group object internal to SimplicialSets, a group action of $\mathcal{G}$ on a simplicial set $X$ is a $\mathcal{G}$-action object internal to SimplicialSets, hence a morphism of simplicial sets of the form

satisfying the action property.

The category of simplicial $\mathcal{G}$-action may be understood as the sSet-enriched functor category from the one-object sSet-category $\mathbf{B}\mathcal{G}$ to sSet:

Properties

Model category structure

(DDK 80, Prop. 2.2. (ii), $\;$ Guillou, Prop. 5.3, $\;$ Goerss & Jardine 09, V Lem. 2.4).

Related concepts

• Borel model structure

• G-set, topological G-space

• ∞-action

References

• Emmanuel Dror, William Dwyer, Daniel Kan, Equivariant maps which are self homotopy equivalences, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 (jstor:2043448)

• Paul Goerss, J. F. Jardine, Section V.2 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)

• Bert Guillou, A short note on models for equivariant homotopy theory, 2006 (pdf, pdf)